Animating indirect lighting
There's currently no simple/automatic option in Blender to have light probes change/bake over time, so you have to do it via scripting.


Select the Scripting workspace
Create a New script
Copy, paste, and modify the below code: change the Path value to wherever you want to output your animation frames, and change the range of frames you want to output. The sample below will save frame 0-29 (does not include 30). 


import bpy

scn = bpy.data.scenes["Scene"]
com = bpy.ops.scene 
path = "d:/blender/output/frame_"
i = 0
for i in range (0,30):
    scn.frame_current = i
    com.light_cache_bake()
    scn.render.filepath = path + str(scn.frame_current)
    bpy.ops.render.render(write_still=True)



Run the script (click Run Script or the play button, depends on version)


Basically what this is doing is going through each frame, performing the "bake indirect lighting" option (the one you manually click under the Render settings), rendering the frame &amp; saving it as frame_0.png, frame_1.png, etc. So it will take a while and there's no "cancel" option, so I'd suggest testing this with a short range of frames to begin with. It may appear as if Blender has crashed, but if you keep an eye on the Path folder, you should see files start to be saved as they render.

When using this method, I only use it for the frames where the lighting changes. I wouldn't want to output the entire animation like this (ie, baking each frame when nothing changes). So I might render frame 0-100 normally, then if my light probe needs to change on frame 101, output frame 101 to 110 with this method, then render the remainder normally &amp; composite everything in After Effects later.

Full credit to the person who came up with this method, as demonstrated here: https://www.youtube.com/watch?v=aAAb6sITkq4

Hope this helps!